Ruelle, Ph.; Thiran, E.; Weyers, J. Implications of an arithmetical symmetry of the commutant for modular invariants. (English) Zbl 1043.81698 Nucl. Phys., B 402, No. 3, 693-708 (1993). Summary: We point out the existence of an arithmetical symmetry for the commutant of the modular matrices S and T. This symmetry holds for all affine simple Lie algebras at all levels and implies the equality of certain coefficients in any modular invariant. Particularizing to SU\((3)_k\), we classify the modular invariant partition functions when \(k+3\) is an integer coprime with 6 and when it is a power of either 2 or 3. Our results imply that no detailed knowledge of the commutant is needed to undertake a classification of all modular invariants. Cited in 10 Documents MSC: 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 11F22 Relationship to Lie algebras and finite simple groups 11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 81R99 Groups and algebras in quantum theory PDFBibTeX XMLCite \textit{Ph. Ruelle} et al., Nucl. Phys., B 402, No. 3, 693--708 (1993; Zbl 1043.81698) Full Text: DOI arXiv References: [1] Moore, G.; Seiberg, N., Nucl. Phys., B313, 16 (1989) [2] Capelli, A.; Itzykson, C.; Zuber, J. B., Commun. Math. Phys., 113, 1 (1987) [3] Ruelle, Ph.; Thiran, E.; Weyers, J., Commun. Math. Phys., 133, 305 (1990) [4] Kac, V., Infinite dimensional Lie algebras (1983), Birkhäuser: Birkhäuser Boston [5] Cardy, J., Nucl. Phys., B270, 186 (1986) [6] Bauer, M.; Itzykson, C., Commun. Math. Phys., 127, 617 (1990) [7] Ruelle, Ph., Ph.D. thesis (September 1990) [8] Altschüler, D.; Lacki, J.; Zaugg, P., Phys. Lett., B205, 281 (1988) [9] Bernard, D., Nucl. Phys., B288, 628 (1987) [10] Christe, P.; Ravanini, F., Int. J. Mod. Phys., A4, 897 (1989) [11] Koblitz, N.; Rohrlich, D., Can. J. Math. XXX, 1183 (1978) [13] Verstegen, D., Nucl. Phys., B346, 349 (1990) [14] Schellekens, A. N.; Yankielowicz, S., Nucl. Phys., B334, 67 (1990) [15] Aldazabal, G.; Allekotte, I.; Font, A.; Núñez, C., Int. J. Mod. Phys., A7, 6273 (1992) [16] Bais, F. A.; Bouwknegt, P. G., Nucl. Phys., B279, 561 (1987) [17] Verstegen, D., Commun. Math. Phys., 137, 567 (1991) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.